Explicit expanders with cutoff phenomena

Eyal Lubetzky; Allan Sly

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arXiv · 1003.3515

Explicit expanders with cutoff phenomena

Eyal Lubetzky·Allan Sly

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Abstract

The cutoff phenomenon describes a sharp transition in the convergence of an ergodic finite Markov chain to equilibrium. Of particular interest is understanding this convergence for the simple random walk on a bounded-degree expander graph. The first example of a family of bounded-degree graphs where the random walk exhibits cutoff in total-variation was provided only very recently, when the authors showed this for a typical random regular graph. However, no example was known for an explicit (deterministic) family of expanders with this phenomenon. Here we construct a family of cubic expanders where the random walk from a worst case initial position exhibits total-variation cutoff. Variants of this construction give cubic expanders without cutoff, as well as cubic graphs with cutoff at any prescribed time-point.

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BibTeX RIS

Eyal Lubetzky, Allan Sly. 2010-03-18. Explicit expanders with cutoff phenomena. https://arxiv.org/abs/1003.3515

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